“The O-Ring Theory of Economic Development,” M. Kremer (1993)

The O-Ring theory is unquestionably one of the most influential papers in applied theory of the past 20 years. The basic idea is simple: Write a Cobb-Douglas utility function where instead of labor entering as a lump of homogenous efficiency units, n units of labor for n tasks must be supplied by n individuals. This represents situations where quantity cannot be substituted for quality. A football team can only have one quarterback, and a kitchen only one head chef. With probability 1-q(i), a given agent fails at his task – all tasks are identical – and destroys 1-q(i) of the final production. This is analagous to the space shuttle Challenger, which exploded despite having only one bad part, the famous O-Ring. For now, assume no uncertainty about skill.

What happens in equilibrium? As in Becker’s marraige model, there is perfect sorting: all workers in a given firm are of the same quality. Each firm makes zero profit by assumption. Since quality enters multiplicatively in the production function, high quality workers are most valuable to firms already employing other high quality workers. The distribution of wages is far more extreme than the distribution of quality, since the marginal product of high quality workers is calculated given their employment at high quality firms. This fact perhaps explains why janitors and secretaries at high productivity firms are paid more than janitors and secretaries at low productivity firms: making a mistake is so costly at the high productivity firms that those companies are willing to pay a very high salary to someone who is even slightly better at their secretary job. The story is less plausible when it concerns janitors – surely interfirm equity issues of some kind provide a better explanation – but the theory is clever nonetheless.

Kremer further considers sequential production. He notes that Rembrandt’s newest assistants prepared the canvas, his advanced assistants painted most of the work, and Rembrandt came in at the end to paint the face and hands. When making a mistake destroys all of the product, the worker least likely to make a mistake performs her task last. If each stage of the sequential production is thought of as a firm who sells their intermediate product onward, the 7th step of production produces profit for the firm of the price of the object sold up to the 8th stage (q) times the probability the 7th step is done correctly (p) minus the price of the good bought from the 6th intermediate company (p’) minus the skill-specific wage. By zero profit condition, this means that the skill specific wage is qp-p’. Since the value of the good increases in every stage, p>p’, and therefore the wage schedule is steeper in skill at higher stages of production. That is, workers who are slightly better get a much higher wage. This is similar in flavor, if not derivation, to Rosen’s tournament model. The argument here ignores the fact that higher-skill workers generally perform more difficult tasks (this point was made to me by a professor here at NW).

Skill can also be endogenized. In the first period, workers choose a level of education which stochastically generates skill. Their true skill is only observed imperfectly by a test score. The worker payoff is her wage minus the cost of her education. Each worker is paid precisely her expected skill. Note the strategic complementarity here: if other workers have a lot of skill, I will also try to get educated, but if other agents do not get much education, then I will not either. These multiple equilibria exist even when looking only at pure strategic symmetric equilibria. Because actual skill level is only observed imperfectly, in a symmetric equilibria, workers end up pooled at firms with other workers at the same test score. Kremer claims this can be a model of statistical discrimination: with multiple equilibria, if firms expect a certain ethnic group to be of low education, then workers in that group will not get a lot of education, confirming the firm hypothesis. Because of the multiplier on wage discussed earlier, the return to education in the low equilibria group will be lower than in the high equilibria group.

The discrimination argument relies heavily on the fact that errors on education and on test scores are normally distributed, hence no matter what the true quality, test scores have full support. If this weren’t the case, a worker in the discriminated group can always get the same amount of education as in the nondiscriminated group, obtain a higher posterior on his test score, and end up matched in the good equilibrium. Essentially, the labor market is not separated. There are multiple equilibria, but there is not necessarily a sorting equilibrium. Indeed, generating a sorting equilibrium would seem to require, for instance, different costs of education.

Today, Kremer is doing a lot of work about replacing (or perhaps reinforcing) the patent system with rewards. That series of papers should be much more influential than they are!